## Introduction

Polybutadiene is a commonly used rubber in many industrial applications, including automotive tires. The mechanical response of the material is qualitatively similar to many other synthetic rubbers and consists of non-linear viscoelasticity with significant strain-rate dependence. In this article I will compare experimental data for a polybutadiene to the best stress-strain predictions of different material models that I calibrated using MCalibration.

## Experimental Data Used for the Polybutadiene Material Model Study

The experimental data that I used for the study is shown in Figure 1. The data consists of compression followed by unloading at 2 different strain rates, and a cyclic compression tests with a mean strain of -25%, a strain amplitude of 1.5%, and a frequency of 1 Hz. 600 cycles were performed until final unloading. Note that this type of cyclic test is not very convenient for material model calibration since the calibration is performed in time-domain and will require about 5,000 time increments to represent that test. In this case there is no significant benefit to having the load cycles compared to a simpler stress relaxation test.

*Figure 1. Experimental data for the polybutadiene. (click on the image for high res).*

Before selecting a material model it is always important to carefully review the experimental data. In this case we can see that the experimental unloading stress is different than the loading stress, indicating that the material is viscoelastic. We can also see that the stiffness is higher at a higher applied strain rate, again indicating that the material is viscoelastic. Another interesting observation is that the initial (small-strain) stiffness of the material is significantly lower than the instantaneous slope of the stress-strain curve right at the start of the unloading. This difference in tangent modulus is a clear indication that the material exhibits Mullins damage during the test. These observations indicate that a viscoelastic material model with Mullins damage will likely be needed to capture the experimentally observed behavior. The results presented below confirm this hypothesis.

## Calibration Results

Direct comparisons between the experimental data and predictions from nine different material models are shown in Figure 2 (click on the individual images). Here are some key observations:

- A hyperelastic model in this case cannot accurately predict the response of the rubber.
- The Ansys TNM is not a good choice for predicting the response of the butadiene rubber.
- The Bergstrom-Boyce (BB) model by itself does not work well, but is reasonably accurate if combined with Mullins damage.
- A hyperelastic model with Mullins damage is as accurate as the Bergstrom-Boyce model with Mullins damage. The optimal calibration of both of these models ignore the strain rate effect.
- A linear viscoelastic model with Mullins damage is slightly more accurate than the BBM model. That is unusual for large strain predictions.
- The Abaqus PRF model (2 Yeoh hyperelastic networks with power flow and Mullins damage) is 22% more accurate than a linear viscoelastic model.
- The PolyUMod TNV model with 2 networks (with Mullins and PSC flow) is 30% more accurate than a linear viscoelastic model.
- The most accurate model I could find is the PolyUMod Parallel Network (PN) model consisting of 2 hyperelastic networks with Mullins damage and a specific flow equation for rubbers that allows for less strain-rate dependence during unloading. That flow equation has id=506 in the PolyUMod User’s Manual. This material model is 48% more accurate than linear viscoelasticity.

*Figure 2. Comparison between experimental data and predictions from 9 different material models.*

*Figure 3. Summary of errors in the model predictions for the 9 material models.*

## Summary

- Hyperelasticity is sometimes a terrible choice for predicting the response of rubbers
- Think carefully about what experimental tests to run
- The PolyUMod TNV and PN models are the most accurate for this polybutadiene